There is a very nice paper
Correlated double-proton transfer. I. Theory
Zorka Smedarchina, Willem Siebrand, and Antonio Fernández-Ramos
It considers an incredibly simple potential energy surface to describe double proton transfer.
x_2 is the corresponding position for the second proton.
The first term describes a quartic potential with an energy barrier for transfer of the proton between the donor and acceptor.
The dimensionless parameter G describes the extent of correlation or coupling between the two hydrogen bonds. The coupling term is chosen to have the important property that it is symmetric in the two co-ordinates but sensitive to their sign. This is an important difference to earlier [rather nice] work by Benderskii et al. who considered competition between two dimensional quantum tunneling paths [instantons] associated with concerted and sequential transfer.
Three types of Potential Energy surface (PES) can occur, depending on the value of G.
The three cases shown below correspond to
a) 0 < G < 1/2
b) 1/2 < G < 1
c) G > 1.
The coordinates x_a=1/2(x_1-x_2) and x_s=1/2(x_1+x_2). The horizontal line between the left and right minimum corresponds to a concerted path.
For the top PES a sequential proton transfer is possible via one of the two intermediate (INT) states.